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(MATH365)[2009](s)midterm~ma_yxf^_10511.pdf
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Midterm Exam MATH 365: Mathematical Biology
J. R. Chasnov
17 April 2009
Answer ALL questions Full mark: 50; .rst and fourth question carries 10 marks; second and third question carries 15 marks. Time allowed C 80 minutes
Directions C This is an open notes examination. You may write on the front and back of the exam papers.
Student Name:
Student Number:
Question No. (mark) Marks
1 (10)
2 (15)
3 (15)
4 (10)
Total

Consider the following model for population growth:
dN
= rN(K . N)(N . M), r> 0, 0 <M <K.
dt
(a)
(2 pts) Determine all equilibrium values of the population N.

(b)
(4 pts) Determine the linear stability of all the equilibria.

(c)
(4 pts) Determine the .nal equilibrium population if (i) 0 <N(0) <M <K; (ii) 0 <M <N(0) <K;


(iii) 0 <M <K<N(0).
Consider a SIR model for an epidemic disease with two types of susceptible individuals, with the .rst type easier to infect than the second. Once infected, there is no di.erence between the two types. Assume a population of initially .N individuals of the .rst susceptible type and (1 . .)N individuals of the second susceptible type.
(a)
(2 pts) Diagram the SIR model for this disease, introducing parameters .1, .2, and ..

(b)
(2 pts) Determine the complete set of di.erential equations.

(c)
(2 pts) Nondimensionalize the equations using N for population size and ..1 for time, and de.ning nondimensional groupings r1 and r2.

(d)
(4 pts) Determine a condition, depending only on the parameters r1, r2 and ., for an epidemic to occur.

(e)
(5 pts) Determine an equation, depending only on the parameters r1, r2 and ., for the fraction of the population that gets sick.

Consider (female) birds that hatch their eggs in late May. A proportion .1 of birds survive their .rst winter and hatch e1 eggs. A proportion .2 of birds that survive their .rst winter also survive their second winter and hatch e2 eggs. No birds survive their third winter.

(a)
(5 pts) Let On and Tn be the number of one-year-old and two-year-old birds in early May at year n. Determine equations for On+1 and Tn+1 in terms of On, Tn, .1, .2, e1 and e2.

(b)
(5 pts) Determine a condition for the bird population to grow.

(c)
(5 pts) If the bird population grows, determine the stable age distribution s = limn.. On/Tn.

Consider a stochastic model for a towns population with constant birth rate b (births per capita per unit time), and constant immigration rate a (immigrants moving to the town per unit time). Neglect the death and emmigration (moving out of town) rates. Assume the population size is initially N0.

(a)
(4 pts) Derive the di.erential equation for the probability pN (t) that the population size is N at time t.

(b)
(4 pts) Derive the di.erential equation for the expected population size .N.(t) from the di.erential equation for pN (t).

(c)
(2 pts) Determine .N.(t).